Equation (REF ) uses the result from [1]}, [2]} that the planetary accretion shock is supercritical and thus isothermal, in the sense that upstream and downstream of the Zel'dovich spike the gas temperature is set by the balance of the incoming flux of material energy and the outgoing radiative flux.
This single temperature \(T_0\)  is shown in Figure REF .
The shock is a “thick–thin” shock in the usual classification (e.g. [3]}): the radiation is diffusive below and free-streaming above the shock.
(For recent discussion of sub- and supercritical shocks, see [4]} and [5]}, [6]}, and importantly [7]} for a brief critical review of inadequate uses of these terms in the literature.) In the limit that the kinetic energy dominates over the internal flux, the shock temperature is [2]}
\( T_0 \approx &~1280~{\rm K}~\frac{1}{f_{\mathrm {fill}}^{1/4}}\left(\frac{R_{\mathrm {p}}}{2~R_{\mathrm {J}}}\right)^{-3/4} \\&\times \left(\frac{\dot{M}}{M_{\mathrm {J}}\,\mathrm {yr}^{-1}}\right)^{1/4} \left(\frac{M_{\mathrm {p}}}{3~M_{\mathrm {J}}}\right)^{1/4}\zeta ^{1/4}.\) 
